Coordinate Geometry: Question 6

Worked solution

Step 1: Recall the distance formula

For two points $(x_1, y_1)$ and $(x_2, y_2)$, the distance between them is

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.$

Step 2: Substitute the coordinates

Take $E(1, 3)$ as $(x_1, y_1)$ and $F(7, 11)$ as $(x_2, y_2)$:

$d = \sqrt{(7 - 1)^2 + (11 - 3)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10.$

Step 3: Check the arithmetic a second time

Recomputing the differences: $\Delta x = 7 - 1 = 6$ and $\Delta y = 11 - 3 = 8$. Since $6, 8, 10$ is a scaled $3$-$4$-$5$ Pythagorean triple ($6=3\times2$, $8=4\times2$, $10=5\times2$), the value $d = 10$ is confirmed without needing a calculator.

Why the other options are wrong

• B ($100$): this is $d^2$, not $d$. The square root at the final step has been forgotten.
• C ($14$): this comes from adding $6 + 8 = 14$ directly, treating the differences like a “Manhattan distance” instead of applying Pythagoras’ theorem.
• D ($8$): this uses only the difference in $y$-coordinates and ignores the $x$-coordinates completely.

Final answer

$\boxed{EF = 10} \quad \text{(Option A)}$